On Achievable Rate Regions of the
Asymmetric AWGN TwoWay Relay Channel
Abstract
This paper investigates the additive white Gaussian noise twoway relay channel, where two users exchange messages through a relay. Asymmetrical channels are considered where the users can transmit data at different rates and at different power levels. We modify and improve existing coding schemes to obtain three new achievable rate regions. Comparing four downlinkoptimal coding schemes, we show that the scheme that gives the best sumrate performance is (i) completedecodeforward, when both users transmit at low signaltonoise ratio (SNR); (ii) functionaldecodeforward with nested lattice codes, when both users transmit at high SNR; (iii) functionaldecodeforward with rate splitting and timedivision multiplexing, when one user transmits at low SNR and another user at medium–high SNR.
I Introduction
We investigate the additive white Gaussian noise twoway relay channel (AWGN TWRC) depicted in Fig. 1. We modify existing coding schemes and obtain three new achievable rate regions. We compare these three modified coding schemes with an existing scheme, and show that different schemes give the best sumrate performance for different SNR regions.
The AWGN TWRC we consider has no direct link between the users; data exchange between the users is done through a relay. The AWGN TWRC is defined by two AWGN channels: the uplink from the users to the relay, and the downlink from the relay to the users.
If we assume that a genie informs the relay of both users’ messages, and only consider the downlink, i.e., how the relay sends these message, the downlink capacity region is known [1, 2]. However, the capacity region of the AWGN TWRC is unknown in general, and the main difficulty lies in determining the best way the relay should process its received signals on the uplink.
Knopp [3] proposed two coding schemes: (i) amplifyforward^{1}^{1}1This scheme was called analog relaying in [3], but is now commonly referred to as amplifyforward. where the relay simply scales its received signals on the uplink and transmits them on the downlink, and (ii) completedecodeforward^{2}^{2}2This scheme was called digital relaying in [3], but is now commonly referred to as decodeforward. We term this scheme CDF to differentiate it from another scheme where the relay only decodes a function of the users’ messages. (CDF) where the relay decodes both users’ messages on the uplink, reencodes and sends both messages on the downlink. Schnurr et al. [4] later proposed the compressforward scheme where the relay quantizes its received signals on the uplink, reencodes the quantized signals, and sends them on the downlink.
CDF, where the relay removes the uplink noise, is downlink optimal in the sense that the downlink channel usage achieves the downlink capacity region^{3}^{3}3Decoding both users’ messages is not always optimal for the uplink. [2]. On the other hand, in the amplifyforward and the compressforward schemes, the uplink channel noise propagates to the downlink and hence they are not downlink optimal. In this paper, we will focus on coding schemes that are downlink optimal.
In CDF, after the relay decodes both users’ messages, instead of sending both messages on the downlink, Kramer and Shamai [1] showed it is also downlink optimal for the relay to transmit only a function of the messages.
Instead of decoding the individual messages and transmitting only a function of the messages, the relay might directly decode this function on the uplink. We term this scheme functionaldecodeforward (FDF). Obviously, the function must be defined such that on the downlink, each user can decode the message of the other user from the function and its own message. In addition, the channel code must match the uplink so that the relay can decode the codeword that carries the function of the users’ messages without needing to decode the messages individually.
For the AWGN TWRC where both users transmit at the same power and at the same data rate, Narayanan et al. [5] proposed FDF using lattice codes^{4}^{4}4Lattice codes have been shown to achieve the capacity of the pointtopoint AWGN channel [6]. (which are linear under the modulolattice operation) where the relay decodes a function, i.e., modulolattice summation, of the user’s messages. This scheme approaches the capacity region of the AWGN TWRC asymptotically as the SNR grows. Using this scheme, both users transmit using the same lattice code and hence at the same rate. For the asymmetrical case where the users transmit at different rates, Knopp [7] proposed a ratesplitting scheme as follows. The user with the lower rate transmits its message using a lattice code. The other user splits its message, and simultaneously transmits the sum (superposition) of (i) the first part of its message using the same lattice code, and (ii) the rest of the message using a random Gaussian code. This scheme introduces interference between the lattice codeword and the Gaussian codeword. To avoid this, Nam et al. [8, 9] used nested lattice codes, where one lattice code is a subset of the other lattice code, so the users transmit at different rates using lattice codes. However, this scheme suffers when there is a large difference between the users’ transmit power levels.
The contributions of this paper are as follows:

We improve the achievable rate region of FDF with nested lattice codes proposed by Nam et al. [9]. We note that for certain SNRs, if a user transmits at a lower (than the maximum allowable) power, the achievable rate of the other user can be increased, and the sum rate can also be increased.

We correct and improve the achievable rate region of FDF with rate splitting and simultaneous transmission proposed by Knopp [7]. When two users transmit using the same lattice code, and the relay decodes the modulolattice addition of the codewords, the achievable rate of used in [7] is incorrect. In addition, similar to FDF with nested lattice codes, we note that using less than the maximum allowable power, the achievable sum rate can be increased.

We propose a coding scheme using FDF with rate splitting and timedivision multiplexing, and obtain a new achievable rate region. With rate splitting, one user transmits using a lattice code, while the other user uses the same lattice code and a Gaussian code. Instead of having the users transmit all codewords simultaneously, we split the transmission of the users into two phases: in the first phase, both users transmit the lattice codewords; in the second phase, one user transmits the Gaussian codeword.

As these schemes have the same downlink performance—all are downlink optimal—we compare their achievable sum rates on the uplink and obtain the following:

In the low SNR region, CDF outperforms the other schemes.

In the high SNR region, FDF with nested lattice codes outperforms the other schemes.

When one user transmits at low SNR and the other user at mediumtohigh SNR, FDF with rate splitting and timedivision multiplexing outperforms the other schemes.

For all SNRs, at least one of the three schemes—(1) CDF, (2) FDF with nested lattice codes, or (3) FDF with rate splitting and timedivision multiplexing—is able to outperform or match FDF with rate splitting and simultaneous transmission.

Ii Channel Model
The AWGN TWRC depicted in Fig. 1 consists of three nodes: nodes 1 and 2 are the users, and node 0 the relay. We define by the transmitted signal of node , and by the received signal of node . The AWGN TWRC is defined by the uplink channel , and the downlink channel , for . Each is subject to the power constraint , and each is independent white Gaussian noise with power , for . We say that users 1 and 2 transmit at SNRs equal to and respectively^{5}^{5}5Different SNRs here can be used to model different channel gains from the users to the relay. Varying and can be used to achieve the same effect on the channels from the relay to the users.. Consider simultaneous uplink and downlink channel uses, in which user 1 is to send an bit message to user 2, and user 2 is to send an bit message to user 1. In the th uplink channel use, each user transmits a function of its message and its previously received signals, i.e., , for all and . In the th downlink channel use, the relay transmits a function of what it previously received, i.e., . After channel uses, user 1 produces an estimate of from its received messages and its own message, . User 2 does likewise to produce . The rate pair is said to be achievable if the probability of decoding error can be made as small as desired, with a sufficiently large . The capacity region is the closure of all achievable rate pairs.
Iii Existing Results
Iiia Capacity Outer Bound
We define . An outer bound to the capacity region of the AWGN TWRC is given as follows:
Theorem 1 ([7])
Consider an AWGN TWRC, a rate pair is achievable only if
(1)  
(2) 
The above outer bound can be obtained from the cutset bound for the general multiterminal network [10, p. 589]. Together, the constraints and give the downlink capacity region, which only depends on the downlink channel parameters , , and .
IiiB CompleteDecodeForward
Using CDF, the following rate region is achievable:
Theorem 2
Consider an AWGN TWRC. CDF achieves the rate pair if
(3)  
(4)  
(5)  
(6)  
(7) 
The above region is obtained by porting the rate region of CDF for the halfduplex discrete memoryless TWRC [3] to the fullduplex AWGN TWRC. Using CDF, the encoding and decoding on the uplink are as follows:
Codelength  

User 1  
User 2  
Relay  decodes and 
where is the length Gaussian codeword transmitted by user , is the random Gaussian code for user with all codeletters independently generated according to the Gaussian distribution with . Here and in the rest of the paper, bold letters denote vectors.
The relay decodes both and on the uplink (i.e., a multipleaccess channel), and it can reliably do so (i.e., with arbitrarily small decoding error) if (3)–(5) are satisfied [10, p. 526]. On the downlink, the relay sends . Knowing its own message, each relay can reliably decode the message of the other user if (6) and (7) are satisfied [11].
Iv FunctionalDecodeForward
Iva FDF with Nested Lattice Codes
We improve on the FDF with nested lattice codes scheme developed by Nam et al. [9] where both users transmit at their maximum allowable power. We modify the scheme such that user transmits at power where , for , and obtain the following new achievable rate region.
Theorem 3
Consider the AWGN TWRC. FDF with nested lattice codes achieves the rate pair if
(8)  
(9)  
(10)  
(11) 
for some . Here .
Note that setting might not give the largest rate region as increasing decreases the RHS of (9), and increasing decreases the RHS of (8).
Without loss of generality, assume that . Using FDF with nested lattice codes, denoted by and , where , the users transmit the following:
Codelength  

User 1  
User 2  
Relay  decodes the function 
where is the length lattice codeword for user , the lattices and satisfy , and are randomly generated length dither vectors which are known to all nodes and are fixed for all transmissions, is the modulolattice operation [6], is a deterministic function of , and . If (8)–(9) are satisfied, the relay can decode reliably [9]. The relay then sends on the downlink. If (10) and (11) are satisfied, both users can reliably decode . User 1 performs to obtain , and user 2 performs to obtain .
IvB FDF with Rate Splitting and Simultaneous Transmission
Next, we correct and modify the achievable rate region using FDF with rate splitting and simultaneous transmission proposed by Knopp [7] to obtain the following new rate region:
Theorem 4
Consider the AWGN TWRC where . FDF with rate splitting and simultaneous transmission achieves the rate pair if
(12)  
(13)  
(14)  
(15) 
for some .
Without loss of generality, assume that . Let where contains bits, and contains bits. Two codes are generated: (i) a lattice code and (ii) a random Gaussian code . The uplink transmissions are as follows:
Codelength  

User 1  
User 2  
Relay  decodes 
and then decodes 
where are length lattice codewords from the same lattice code, is the length Gaussian codeword, and and are randomly generated length dither vectors which are known to all nodes and are fixed for all transmissions.
The relay first decodes by treating as noise, subtracts off its received signals, and then decodes . The relay then sends on the downlink. The above scheme was proposed in [7]. We make the following modifications:

We note that the users might not transmit at their full available power, as the power used by user 1 to transmit acts as an interference when the relay decodes . We propose that both users use to transmit . User 1 then uses a fraction of its remaining power of to transmit .

We correct a minor error in the rate region for reported in [7] (c.f. (12)). is the rate of the lattice code used by both users with the same power. The relay attempts to decode the modulolattice sum of the lattice codewords, i.e., , in the presence of channel noise of power and of power . It has been shown in [5, 12] that the relay can reliably decode the modulo sum of lattice codewords if , where the SNR in our modified scheme is . Narayanan et al. [5] conjectured that the rate of (reported in [7]) cannot in fact be achieved.
The relay can reliably decode if (12) is satisfied. The relay then removes and can reliably decode if (13) is satisfied. The relay then sends on the downlink. Knowing , user 1 removes from its received signals, and it can decode if (15) is satisfied. It obtains from and . If (14) is satisfied, user 2 can decode , from which it can obtain .
Remark 1
It is also possible to decode the Gaussian codeword first by treating the lattice codewords as noise. However, deriving the rate expression for this scheme is difficult as the effective noise in this case is the sum of Gaussian noise and lattices. Similar difficulty is encountered when one attempts to derive the rate expression for simultaneous decoding at the relay, as lattice decoding (using ML decoding) is used to decode the modulolattice sum of the lattice codewords and typical set decoding is used to decode the Gaussian codeword.
Remark 2
This uplink scheme where a user simultaneously transmits lattice and Gaussian codes was also considered by Baik and Chung [13]. However, they employed a different coding scheme on the downlink, i.e., the relay’s encoding.
IvC FDF with Rate Splitting and TimeDivision Multiplexing
Next, we propose another coding scheme by modifying the rate splitting scheme in Sec. IVB, and obtain the following:
Theorem 5
Consider the AWGN TWRC where . FDF with rate splitting and timedivision multiplexing achieves the rate pair if
(16)  
(17)  
(18)  
(19) 
for some .
Again, we assume that , and we split the message , where has bits, and has bits. We generate two codes: (i) a lattice code with codewords of length each, and (ii) a random Gaussian code with codewords of length each. In the first uplink channel uses, both users transmit using the same lattice code with power . In the next channel uses, user 1 transmits with the Gaussian code using its remaining power . The uplink encoding and decoding are as follows:
Codelength  

User 1  
User 2  
Relay  decodes  decodes 
where are length lattice codewords from the same lattice code, is the length Gaussian codeword, and and are randomly generated length dither vectors which are known to all nodes and are fixed for all transmissions.
In the first channel uses, the relay decodes the summation of lattice codewords . The relay can reliably decode if (16) is satisfied [5]. The next channel uses are AWGN pointtopoint channel uses from user 1 to the relay without user 2’s interference. So, the relay can reliably decode if (17) is satisfied. After the relay obtains , the downlink transmission is the same as that of FDF with rate splitting and simultaneous transmission. Hence, we get (18)–(19).
Remark 3
Our proposed scheme differs from FDF with rate splitting and simultaneous transmission in (at least) the following two ways:

The lattice codes and the random Gaussian codes are of different lengths. The codelengths are proportional to the time fraction of the respective transmissions.

There is no interference between the lattice codewords and the Gaussian codeword.
Remark 4
The downlink constraints on the achievable rate regions of all four schemes discussed coincide with the downlink capacity region. So, these schemes are downlink optimal.
V Sum Rate Comparison
In this section, we compare the sum rate of the four schemes described in the previous sections to the sum rate upper bound. Note that a larger sum rate does not necessarily mean the entire twodimensional rate region is larger. We fix the uplink channel noise power . Without loss of generality, we assume that . For the case of , we simply reverse the roles of the users.
As these four schemes are downlink optimal, we only compare their uplink performance. This can be done by setting the relay power to be sufficiently high such that the downlink constraints will always be satisfied.
We fix and plot the maximum achievable by each scheme by varying . In Fig. 2, we constrain user 2 to transmit at low SNR. When the other user (user 1) also transmits at low SNR, CDF gives the best performance. Still keeping user 2’s SNR low, when user 1 transmits at high SNR, FDF with rate splitting and timedivision multiplexing outperforms the other schemes. From Fig. 3, when both users are transmitting at high SNR, FDF using nested lattice codes outperforms the other schemes.
Fig. 4 gives a summary of schemes that achieve the highest sum rate for different and normalized by . We observe the following: (i) When both users transmit at high SNR, FDF with nested lattice codes outperforms other schemes. (ii) When both users transmit at low SNR, CDF is the preferred scheme. (iii) When one user transmits at a low SNR and the other user at mediumtohigh SNR, FDF with rate splitting and timedivision multiplexing achieves the highest sum rate. In the equalSNR region where the three FDF schemes attain the highest sum rate (i.e., ), the three schemes effectively reduce to the same scheme: both users transmit using only a lattice code (the same lattice code) at the same power and no rate splitting is done. Also seen from the figure, FDF with rate splitting and simultaneous transmission does not give the (strictly) best sum rate at any SNR, i.e., one of the other schemes can always outperform or match it.
Using CDF, the relay needs to decode both the users’ messages, c.f. (5). Because of the concavity of the function, this constraint limits its performance at mediumtohigh SNR. FDF, on the other hand, does not suffer from this problem. However, because of the modulolattice operation (see [14] for more discussion), FDF (which uses lattice codes) achieves rates up to where , while CDF (which uses Gaussian codes) achieves rate up to . So, FDFbased schemes do not perform well at low SNR. This explains why CDF performs better at low SNR, while FDFbased schemes perform better at mediumtohigh SNR.
Using FDF with nested lattice codes, lattices of two different sizes (which depend on the transmit power) are used in the transmissions of the two users. As the relay decodes the modulosum of transmitted codewords with respect to the bigger lattice, the rate of the user that transmits using the smaller lattice (lower transmit power) is penalized. So, when one user transmits at high power and the other user at low power, the sum rates of FDF with nested lattice codes are affected; the sum rates of CDF are also affected by the reason given in the previous paragraph. In this region, our proposed FDF with rate splitting and timedivision multiplexing is able to give better sumrates, as it does not suffer from the problems of the need to decode both users’ messages and mismatched lattices.
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